![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > psubclssatN | Structured version Visualization version GIF version |
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclssat.a | β’ π΄ = (AtomsβπΎ) |
psubclssat.c | β’ πΆ = (PSubClβπΎ) |
Ref | Expression |
---|---|
psubclssatN | β’ ((πΎ β π· β§ π β πΆ) β π β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psubclssat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
2 | eqid 2733 | . . 3 β’ (β₯πβπΎ) = (β₯πβπΎ) | |
3 | psubclssat.c | . . 3 β’ πΆ = (PSubClβπΎ) | |
4 | 1, 2, 3 | psubcliN 38809 | . 2 β’ ((πΎ β π· β§ π β πΆ) β (π β π΄ β§ ((β₯πβπΎ)β((β₯πβπΎ)βπ)) = π)) |
5 | 4 | simpld 496 | 1 β’ ((πΎ β π· β§ π β πΆ) β π β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3949 βcfv 6544 Atomscatm 38133 β₯πcpolN 38773 PSubClcpscN 38805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-psubclN 38806 |
This theorem is referenced by: pmapidclN 38813 psubclinN 38819 paddatclN 38820 pclfinclN 38821 poml6N 38826 osumcllem3N 38829 osumcllem9N 38835 osumcllem11N 38837 osumclN 38838 |
Copyright terms: Public domain | W3C validator |